Theory for Anomalous Electric Double-Layer Dynamics in Ionic

Apr 7, 2014 - Phone: 91-11-27666646-188, 149. ... Comparisons of the theoretical impedance and capacitance responses with recent experiments on ...
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Theory for Anomalous Electric Double-Layer Dynamics in Ionic Liquids Maibam Birla Singh and Rama Kant* Department of Chemistry, University of Delhi, Delhi 110007, India ABSTRACT: Anomalous slow dynamics with impedance phase modulation and two capacitive arcs are seen in the recent experiments with room-temperature ionic liquid (RTIL) on gold single-crystal electrodes. Single-crystal electrodes have low surface atomic heterogeneity and can show crystal face dependent multiorientation adsorbing states. We have extended our recently developed model of electric double-layer (EDL) impedance of a heterogeneous electrode with single state compact layer (see J. Electroanal. Chem. 2013, 704, 197−207) to a multistate compact layer for accounting ion shape asymmetry in RTILs. The modified multistate model incorporates ion shape asymmetry dependent molecular properties and related relaxation dynamics in the compact layer. Our model with two or three ion orientation states in the compact layer for systems with shape asymmetric cations explains such anomalous dynamics. Comparisons of the theoretical impedance and capacitance responses with recent experiments on electrode/IL systems are in good agreement.

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INTRODUCTION Room-temperature ionic liquids (RTILs) have attracted considerable research interest in electrochemistry due to their potential applications in electrochemical supercapacitors,1,2 field-effect double-layer transistors,3 dye-sensitized solar cells,4,5 and fuel cells.4,5 This is due to the high ionic conductivity, nonvolatility, low vapor pressure, thermal stability, and wide electrochemical window that ionic liquids possess.1−7 A general but important feature of ILs is the size and shape difference of cations and anions. Usually the size of a cation is larger due to the presence of ring structure. This feature of cations results in asymmetry in size, shape, and effect of the nature of the structure of the EDL formed at the electrode/IL interface.8 An interesting question is how the shape and size factors are going to affect the dynamics of the EDL. Recent experimental,9−11,18 theoretical,12,13 and simulation14−17 studies at the electrode/IL interface show that the potential variation leads to restructuring of the Helmholtz layer in the interface, resulting in potential-dependent capacitance and its dynamic response. Various other factors like surface roughness or disorder,19−21 temperature of interface,22−25 or chemical nature (π- or non-π-system)26,27 of the cation and nature of the electrode (metal, semimetal, or semiconducting)27,28 are found to affect the capacitance and dynamics of the IL interface. While theories for the diffuse layer12,13,29,30 are developed, accounting for electrostatic correlations, steric, and molecular size asymmetry effects, no proper attention is paid to the compact layer part of the electric double layer. The recent experiments on electrode/IL interface using AFM,31 STM,31,32 and spectroscopy9,33−35 suggest that it is the compact part of the EDL nearest to an electrode which is strongly affected by change in potential, temperature, or ion size asymmetry. In the case of a highly concentrated ionic system like ionic liquids, the Debye screening length is small (usually a fraction of © 2014 American Chemical Society

nanometers), thus the diffuse layer thickness is comparable to the size of the ions and the thickness of the Helmholtz layer organization. For a pure ionic liquid with no solvent, the ion concentration is essentially the same throughout the system, and a difference in structure can only arise in the Helmholtz layer. However, the role of the compact layer is not yet clearly identified theoretically as highlighted in experiments. An important quantity which gives the information about the structure of the electrode/IL interface and dynamic process at the interface is the characteristic time or frequency of relaxation. This time can be obtained from electrochemical impedance spectroscopy (EIS) and can be used as a fingerprint parameter to distinguish the nature of different processes at the EDL. Recently obtained EIS experimental data by Pajkossy et al.36,37 for an electrochemical interface between 1-butyl-3methyl-imidazolium hexafluorophosphate, BMI PF6, with Au(111) and Au(100) show two interesting features: (i) oscillation in phase value in low frequencies which look like camel humps in shape and (ii) change in the position of the phase peak. The EIS spectra suggest kinetics of anion/cation replacement in the compact (innermost) part of the double layer. A similar observation was observed in interfacial capacitance data obtained by Roling et al. for the electrochemical interface between single crystalline Au(111) and 1butyl-1-methylpyrrolidinium tris(pentafluoroethyl)trifluorophosphate, [Py1,4][FAP],25,38 and 1-ethyl-3-methylimidazolium tris(pentafluoroethyl)trifluorophosphate, [EMIm][FAP].39 These observations revealed the existence of two or more cation orientation-dependent relaxation processes, a fast process taking place on a scale of milliseconds. Received: January 13, 2014 Revised: April 4, 2014 Published: April 7, 2014 8766

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Figure 1. Schematic illustration of EDL at a heterogeneous electrode/IL interface showing orientations of ions in the Helmholtz layer (adapted from ref 9).

a slow one in time of seconds, and maybe an intermediate time relaxation. These time scales are related to the structural reorganization of the ion and/or charge redistribution in the innermost ion layer. While Pajkossy et al. use both R−C− CPE36 and R−W−C37 equivalent circuit models, Roling et al. prefer the Cole−Cole25,38,39 equivalent circuit model to calculate the time scales emerging out of EDL reorganization. There is no agreement at present, in the literature, on an impedance model of the IL/electrode which can describe the dynamics of ILs. To establish a correct premise of the experimental understanding and to investigate how ion spatial orientation affects the dynamics of the EDL are very important. In this article we report a model of the electrode/IL double layer based on a recently developed model of EDL dynamics on heterogeneous40 and rough electrodes (Figure 1).41 Our model40,41 was successful in describing the dynamics of Hg/glycerol, Hg/ glycerol−water, single and polycrystalline Au in HClO4, and rough Pt electrodes in H2SO4. Here we further generalize our model for the case where ions can exist in more than one adsorbed state in the Helmholtz layer (HL) with different characteristic relaxation times. We assume that the diffuse layer is described by fast Debye−Falkenhagen dynamics.42 As a special case of the generalized HL model, a two or more adsorbed states model due to ion orientation asymmetry near a planar electrode with heterogeneity is developed. The objective of the paper is to understand the interplay between the various adsorbed states arising in the compact layer and their time scales due to different orientations of adsorbed ions (cation or anion) of IL and the broadening of relaxation time scales due to residual heterogeneity of the surface.

κ −1 =

THEORETICAL METHODS EDL Dynamics with Multistate Ion Adsorption in the Compact Layer. The dynamics of the diffuse part of the double layer can be described by the linearized Debye− Falkenhagen (LDF) equation for potential ϕ with respect to bulk solution40,41 2

∂ϕ/∂t = D(∇ − κ )ϕ

2NAe 2I

(2)

Here I = (1/2)∑iciz2i is the ionic strength; kB is the Boltzmann’s constant; NA is the Avogadro number; T is the temperature; ϵ0 is the permittivity of free space; and ϵr is the dielectric constant of ionic liquid. The current density (j) crossing at the outer Helmholtz plane (OHP) from solution is given by Ohm’s law, j = −σ ∇ϕ, where σ is the interfacial conductivity of IL at the OHP. This current arriving at the OHP (at z = 0 plane) should be equal to the product of potential drop at the OHP from the electrode (ϕ̃ − ϕ) to the admittance (yH) of the Helmholtz layer (see eq 6); i.e., (ϕ̃ − ϕ)yH where ϕ̃ is the potential of the electrode and ϕ is the potential at and beyond the OHP. Now, equating the current on both sides of the Helmholtz layer we have the Robin (or impedance) boundary condition:40,43−48 −σ(∂ϕ/∂z)|z=0 = yH̅ (ω)(ϕ̃ − ϕ (z = 0)) where yH̅ (ω) is the admittance of the Helmholtz layer in the heterogeneous electrode (the two- and three-state compact layer models will be discussed in the following Discussions section). The impedance boundary introduces a fundamental phenomenological length40 |σ/yH̅ (ω)| which is the equilibration length of the EDL (see the Results and Discussion for the physical significance of this length). Usually the geometric separation length L in between the working and counter electrodes is much larger than the Debye length (κ−1) (so there are no overlaps of the two diffuse layers), hence it is appropriate to use (in most cases) the semi-infinite (bulk) boundary condition, ϕ(∞) = 0. Our model essentially assumes that the surface heterogeneity mainly influences the compact layer dynamics, while the diffuse double-layer (with relatively weak electric field region compared to a compact layer) dynamics is governed by the (linearized) Debye−Falkenhagen equation for the potential. The total impedance is obtained as the sum of the Helmholtz layer and diffuse layer impedance40 as

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2

ϵrϵ0kBT

Z(ω) = Z H(ω) + ZG(ω)

(3)

The dynamic impedance response is series addition of compact layer impedance

(1)

Z H(ω) = 1/(A 0yH̅ (ω))

where D is the diffusion coefficient and κ−1 is the Debye screening length

(4)

and the diffuse layer impedance 8767

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Z0 1 = 1/2 A 0σκ 1 + iωτD A 0D 1/τD + iω

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yH̅ (k) (ω) =

(5)

where A0 is the area of the electrode, Z0 = (D/σ), and τD = 1/ κ2D. The admittance of the HL, yH̅ (ω), with surface heterogeneity exponent, γ, is expressed as40,49 yH̅ (ω) =

(9)

This model can be used for identifying two extreme characteristic frequencies in the anomalous CPE regime of experimental data, viz., small and large characteristic frequencies. These frequencies may represent two idealized states: vertical and horizontal orientation of the cation on the surface of the electrode. For such a two-state system (k = I and II), the small relaxation time τIH = RIHcIH gives the time required to achieve vertical configurations of ions, whereas τIIH = RIIHcIIH may give the time required to achieved horizontal configuration. The time of reorganization of ions in HL will depend on the effective resistances and capacitances related to the configurational states of ions in the compact layer. However, a third intermediate state may exist in solvent-free ionic liquids and can be modeled between these two extreme states. In the presence of solvent or mixed ILs, several intermediate states may appear to generate an effective suppressed phase angle behavior in the CPE regime.

1 RH ⎡ ⎤ 1 ⎢1 − ⎥ 1 + RHR ct−1 + (iωτH0)γ (1 + RHR ct−1)1 − γ ⎦ ⎣ (6)

where RH is the HL resistance; Rct is the charge transfer resistance; and τ0H = RHcav H is the site heterogeneity averaged relaxation time of HL with symmetrical ions. The site-averaged net compact layer capacitance cav H accounts for the different capacitance contributions of all surface sites. Here the compact layer capacitance is given by a classical Helmholtz model and cav H = ϵHϵ0/rH where ϵH is the dielectric constant of the compact layer, ϵ0 the permittivity of free space, and rH the thickness of the compact layer. When Rct → ∞, eq 6 reduces to purely nonFaradaic (blocking) interface with admittance density yH̅ (ω) = (RH)−1[1 − {1 + (iωτH0)γ }−1]

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RESULTS AND DISCUSSION The phenomenological equilibration length |σ/yH̅ (ω)| essentially governs interplay of the local dynamics of compact and diffuse parts of the double layer. Its frequency-dependent behavior suggests the extent of influence in the interfacial region by the change in relaxation times due to capacitive and resistive behavior of adsorbing ions in the compact layer. For a two-state model whose impedance is given by eq 8 we can identify two frequency regimes characterized by ωIH = 1/(RIHcIH) and ωIIH = 1/(RIIHcIIH). The number of layers of ions may be influenced by the structural reorganization of ions in compact EDL and has the value |σ/(yH̅ (ω)dm)| (where dm is the molecular diameter of the ion) at a given frequency. For a typical ionic liquid with dm = 0.84 nm (here corresponding to a thickness of molecular layer in the [C8C1Im][Tf2N]/Au(111) interface51), τIH = 60 ms, τIIH = 1.2 ms, ϵH = 11.452 (typical range of dielectric value found in ILs), γ = 0.9, σ = 1.46 mS/cm36 (typical order), A0 = 1 cm2, and w1 = 0.5. At frequency ω < ωIIH the value of |σ/(yH̅ (ω)dm)| is very large, suggesting that the reorganization of ionic liquid ions can take place involving many molecular layers. This sluggish process allows the dynamics equilibration between the many layers of ions of IL in bulk. This regime is also the CPE regime (see discussion of impedance plots for a clearer understanding). At ωIIH, the value of |σ/(yH̅ (ω)dm)| ∼ 7 corresponds to a thickness of ∼6 nm which is in the typical order obtained from experiments too.51 This suggests that the influence in diffuse layer and bulk will be seen only up to 7 molecular layers from the electrode surface. At ωIH, the |σ/(yH̅ (ω)dm)| ∼ 1, suggesting the dynamic changes to the local kinetic hopping relaxation with rate constant equal to κ2D. Here the ion hopping motion in the diffuse layer causes relaxation dynamics. This relaxation process is mainly controlled by molecular dimensions which is comparable to diffuse layer thickness in solvent-free IL. Thus, in general the reorganization dynamics take place when the frequency is ωIIH < ω < ωIH and hopping relaxation related to fast relaxation is ωIH ∼ κ2D.

(7)

Dispersion in relaxation time may result from surface factors like atomic-scale inhomogeneities, polycrystallinity, grain boundaries, defects, or electrode roughness. It is known that different crystal faces have different site-dependent capacitance,50 and this will influence the characteristic time of relaxation of HL. Apart from the electrode surface factors, different states in the ions of IL can exist because of different ion spatial configurations due to its shape asymmetry. This is particularly important where the asymmetry in shape and size of ions of IL is large. STM31,32 and spectroscopic33−35 studies show that the cation-rich compact layer has electrode potential or temperature-dependent structures with nonuniform distribution of various orientation states of ions. Shapes and configurational complexity of the RTIL cation cause an orientation parallel or (tilted) normal to the surface. Change in orientation moves the positive charge headgroup close to or away from the electrode which causes the existence of different capacitive processes, their characteristic relaxation time scales, and weight of each state in the compact double layer. STM studies show that the cation-rich compact layer has electrode potential-dependent structures. These potential-dependent compact layer structures can be looked upon as changing the distribution of various orientation states of asymmetric ions. Hence, the weight of each state in our theory will depend upon potential, temperature, and crystal face of the systems. However, in the presence of more than one adsorbing state in the compact layer due to ion shape and size asymmetry as in IL, the admittance of the electrode/IL interface may be expressed as yH̅ (ω) =

∑ wkyH̅ (k) (ω) k

1 RHk ⎡ ⎤ 1 ⎢1 − ⎥ ⎢⎣ 1 + RHk R ct−1 + (iωτHk )γ (1 + RHk R ct−1)1 − γ ⎥⎦

(8)

where wk is the surface occupancy fraction of the k-th state and its admittance is given by 8768

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Effect of Distribution of Two States in the Compact Layer. It is well-known from experiments38 that the innermost layer in the IL/electrode is usually spatially organized with preferred orientation due to electrostatic and excluded volume interactions. Whereas the next layer or the so-called “transition zone” reflects a partial behavior of ion ordering due to the inner layer, at the same time the structure of this layer is influenced by the molecular flexibility.25,53 Different states of compact layer may arise due to application of potential, size, and shape asymmetry or due to different orientations of these ions. This will result in multiple contributions to dynamics. The weighting fraction wk represents the various contributions from molecular states existing in the compact layer. Figure 2 shows the comparison of dynamics in terms of (a) magnitude of impedance, (b) phase of impedance, and (c) complex capacitance of the one-state with two-states (obtained from eq 8) model with change in weighting fraction w1 and w2 = 1 − w1. The capacitance in states I and II is assumed to be equal; i.e., the ratio of capacitance γc = 1. In the plots the red (dotted) and blue (dashed) lines correspond to one-state models with relaxation times, τ, equal to 44 and 0.59 ms, respectively. The plots were generated for A0 = 0.38 cm2 and Rct = 20 MΩ cm2. The black (dashed, solid, and dotted) lines represent the effect of change in w1 (= 0.2, 0.5, and 0.7) in the two-state model with the slowest and fastest relaxation times of 44 and 0.59 ms, respectively. The impedance plots shown in (a) show that all responses of states, one and two (with fractions of states), merge when ω < ωIIH. This indicates a configurationaveraged behavior due to multiple switching between states resulting in a common indistinguishable response. At frequency ωIIH < ω < ωIH the impedance plots show a change in the slope due to change in w1. In this frequency regime, we can distinguish the possible states due to ion reorganization. This is also seen as a camel hump shape in phase plot (b). At ω > ωIH, the value of impedance shows a constant value indicating a kinetic controlled regime. Thus, the effect of w1 variation is seen in ω > ωIIH and results in impedance values with different magnitude. The phase behavior of two states is represented by black (dotted, solid, and dashed) lines shown in (b), and it clearly indicates that the value of phase is less than 90° in the sluggish CPE regime. At ωIIH < ω < ωIH the phase decreases from a plateau CPE value and again rises, showing a hump whose phase value is always less than the CPE phase value. This gives rise to a phase plot which looks like a camel hump in shape. The origin of such a shape in phase plots is due to the existence of more than one orientational state but not due to CPE.36,37 Such phase behavior is seen in the recent EIS measurement in the electrode/IL interface.36,37 As the fraction of one state increases over the other by changing the value of w1, the phase behavior gets biased toward one state whose contribution is greater than the other one. However, the magnitude of the phase value at both the peak and valley is decreased with increasing w1. The change in w1 has little effect on the value of frequency of relaxation ωHII and ωHI suggesting that the reorganization of the interface is weakly affected by the weighting fraction of different states present in the Helmholtz layer. At ω > ωIH the phase value is zero, which indicates the onset of kinetic hopping of ions in the diffuse layer for relaxation. The complex capacitance plots in terms of real, C′, and imaginary, C″, capacitance in (c) shows one capacitive arc for the one-state (red dotted and blue dashed lines) model and two

Figure 2. Effect of different fraction of two states of ions in the compact layer on (a) magnitude of impedance, (b) phase, and (c) complex capacitance. The frequency ν (Hz) is related to angular frequency ω (radians per second) as ν = ω/2π. The plots were generated with physical parameters: ϵH = 11.4, rH = 0.35 nm, γ = 0.94, γc = 1. In the plots the red (dotted) and blue (dashed) lines correspond to one-state models with relaxation times τ equal to 44 and 0.59 ms, respectively. The plots were generated for A0 = 0.38 cm2 and Rct = 20 MΩ cm2. The black (dashed, solid, and dotted) lines represent the effect of change in w1 (= 0.2, 0.5, and 0.7) in the twostate model with the slowest and fastest relaxation times of 44 and 0.59 ms, respectively.

capacitive arcs for the two-state (blackdotted, solid, and dashed lines) model. The two arcs represent that there are two relaxation processes taking place simultaneously but in two different frequency scales. The weight of various states (wk) has a significant influence on the size of the two arcs and changes of the capacitance value at frequencies ω < ωIH, i.e., both at CPE and ion reorientation frequencies. The value of wk and its relative contribution will either increase or decrease the size of the arc. Effect of Electrode Heterogeneity. Figure 3 shows the effects of electrode heterogeneity on (a) magnitude of impedance, (b) phase angle of impedance, and (c) complex capacitance. As we increase the heterogeneity (by decreasing the value of γ), there is a decrease in the slope of impedance plots in the CPE regime at ω < ωIIH. The phase plots (b) show 8769

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reorganization regime. The imaginary capacitance C″ also shows an increase with increasing γ, thus the energy losses increase due to disorder of the surface. Effect of Compact Layer Resistance Change Due to Orientation. Figure 4 shows the effect of relative change in

Figure 3. Effect of electrode heterogeneity on the existence of different orientational states of ions in the two-state compact layer (a) impedance, (b) phase, and (c) complex capacitance. The value of γ is varied as 0.9 (red), 0.96 (blue), and 0.99 (black). The plots were generated with physical parameters: ϵH = 11.4, rH = 0.35 nm, w1 = 0.7, and w2 = 0.3.

an increase in heterogeneity curtails the features (distinct two humps) to an averaged out feature with little or indistinguishable states in the CPE regime. This can be understood as, by increasing the heterogeneity of the surface the surface disorder feature is templated in the interfacial layers. Thus, a distinction of vertical or horizontal orientational states is not clear. In fact, the heterogeneity of the surface destroys the spatially organized ion orientation found in the pure perfect single crystal. An important feature is the location of a phase hump at a particular frequency. This is insensitive to changes in the heterogeneity indicating that the reorganization of ions from vertical to horizontal or vice versa takes place at the same rate. However, at higher frequencies, ω > ωIH, the interface constitutes all the possible states similar to bulk IL. Here the phase value goes to zero, suggesting the resistive nature of the interface due to the onset of kinetic hopping of ions from the polarized layer to the bulk. The complex capacitance plots in Figure 3(c) show that both the low frequency (sluggish CPE) and intermediate (ion restructuring) regimes are strongly affected by heterogeneity. Large heterogeneity results in an increase in effective CPE capacitance with a simultaneous decrease in C′ at the

Figure 4. Effect of resistance change due to different orientational configuration of the ion in the compact layer on (a) impedance, (b) phase, and (c) complex capacitance. The value of γR is varied as 1 (black), 10 (brown), 100 (blue), and 1000 (red). The plots were generated with physical parameters: ϵH = 11.4, rH = 0.35 nm, w1 = 0.7, γ = 0.95, RIH = 10 Ω cm2, and T = 293 K.

resistance due to orientation which is characterized by the ratio of the resistance of the two different orientations γR = RIIH/RIH in the compact layer on (a) the magnitude of impedance, (b) the phase angle of impedance, and (c) the complex capacitance. At ω < ωIIH, all the impedance response merges with a common slope corresponding to the CPE behavior. At ωIIH < ω < ωIH (ion reorganization regime) there is a difference in the slope of impedance plots due to change in γR. At ω > ωIH the ion reorganization to a vertical orientation is complete, and the process of ion hopping from the polarized layer starts. This shows a constant resistive behavior in impedance plots. 8770

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in a couple of decades in frequency which is more clearly observed in the phase plots in Figure 5(b). However, unlike in the case of change in γR (see Figure 4(b)), the change in γc has little effect on the characteristic frequency of the phase hump. Figure 5(c) shows that with increasing γc only the capacitive arc in low frequency at ω < ωIH is affected. The high-frequency capacitive arc is insensitive to capacitive changes due to changes in the orientation of ions. Figure 6 shows the comparison of one, two, and three states in (a) the magnitude of impedance and (b) the phase of the

The phase plots in (b) show a clear distinction of existence of two states. The increase in γR shows the emergence of a camel hump shape phase indicating that the time of relaxation between the slowest and fastest process is large. There is a change in the characteristic frequency at which the hump is observed with further delay in the onset of the CPE regime at ω < ωIIH. The distinction of two states is also indicated by the two capacitive arcs with an increase in γR in complex capacitance plots in (c). However, this feature is indistinguishable when the characteristic frequencies of two processes are comparable. Effect of Relative Capacitance Change Due to Two Orientations. Due to the different potential-dependent orientation of ions the distance of closest approach of ions can change due to the difference in size along the long and short dimension. Figure 5 shows the effect of relative change γc

Figure 6. Effect of fraction of states on (a) magnitude impedance and (b) phase response of intermediate states in the three-state model. The dotted (black, brown, and orange) lines represent plots of the onestate model whose relaxation times τ are 0.61 ms, 61 ms, and 0.55 s, respectively. The dotted magenta line corresponds to the two-state model with relaxation times of 0.61 ms and 0.47 s with state fractions of 0.1 and 0.9, respectively. The solid black, blue, and red lines show the effect of variation of the fraction of the intermediate (second) state in a three-state model. Figure 5. Effect of capacitance change due to different orientational configurations of ions in the compact layer on (a) impedance, (b) phase, and (c) complex capacitance. The value of γc is varied as 0.1 (red), 0.3 (blue), 0.7 (brown), and 1 (black).

impedance. The plots were made using HL admittance given by eq 8. The dotted (black, brown, and orange) lines represent plots of one-state models whose relaxation times τ are 0.61 ms, 61 ms, and 0.55 s, respectively. The dotted magenta line corresponds to the two-state model with relaxation times of 0.61 ms and 0.47 s with state fractions 0.1 and 0.9, respectively. The solid black, blue, and red lines show the effect of variation of the fraction of the intermediate (second) state in a threestate model. In the plots the contribution from the state with relaxation time of 0.61 ms was fixed with w1 = 0.1. The contribution from states with relaxation times of 61 ms and

= cIIH/cIH which accounts for the capacitance changes due to ion orientation. The plots were generated with physical parameters: ϵH = 11.4, rH = 0.35 nm, w1 = 0.5, γ = 0.95, γR = 100, and RIH = 10 Ω cm2. In Figure 5(a), on increasing γc, only the magnitude of impedance at ω < ωIH is affected. The capacitive change due to the orientation of ions affects the slow dynamics at lowfrequency CPE regimes. This change in capacitance can be seen 8771

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with different relaxation time scales. The red lines in Figure 7(a) and (b) represent the magnitude and phase of the impedance response of the three-state model. We found that the three-state model gives a better fit than the two-state model. The parameters used to fit the plots are listed in Table 1. Figure 8 shows the complex capacitance theoretical plots along with the experimental data of Au(111) in [EMIm]-

0.55 s was varied as (w2,w3) = (0.1, 0.8), (0.2, 0.7), (0.3, 6). All the plots were generated with the following physical parameters: ϵH = 11. 7, rH = 0.285 nm, γ = 0.95, γc = 1, and A0 = 1 cm2. The change in the fraction of contribution of the intermediate state shows a broadening of the camel hump shape phase response with simultaneous increase in the phase peak as its contribution is increased.

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EXPERIMENTAL COMPARISON Figure 7 shows the comparison of the experimental magnitude of the impedance (a) and the phase of impedance (b) data of

Figure 8. Comparison of two-state model capacitance spectra curves with experimental capacitance data of Au(111) in [Py1,4][FAP] (brown square)25 and [EMIm][FAP] (black circle).39

[FAP]39(black circle) and [Py1,4][FAP]25 (brown square). We employ a two-state model for the theoretical curves. The physical parameters for the theoretical curves are listed in Table 1. In agreement with the theory, the experiment shows the presence of two states indicated by two capacitive arcs. The data use the same Au(111) single-crystal electrode but differ in nature of cations, i.e., [EMIm]+ and [Py1,4]+. These cations differ in both physical size and chemical properties. [EMIm]+ is a π-system, and [Py1,4]+ is a non-π system. Table 1 consists of the values of physical parameter used to generate the theoretical plots for comparing with the experimental data. The static dielectric constant value of [EMIm][FAP], i.e., 12.8,52 was taken as the dielectric constant and molecular radius of [EMIm]+; i.e., 0.26 nm54 was taken as the thickness of HL and was used to fit the data of Au(111) in [EMIm][FAP] (black circle). Similarly ϵH = 11.4 and rH = 0.29 nm (corresponding to the radius of [Py1,4]+)54 were used to fit the Au(111)/[Py1,4][FAP] data. A potential of −0.9 V versus Fc0/Fc+ was applied between [Py1,4][FAP] and the Au(111) interface, whereas the measurement of complex capacitance between [EMIm][FAP] and the Au(111) interface was done in 0.0 V. One of the basic differences of ILs used in experiments is the nature of the cation. The cations [BMI]+, [EMIm]+, and [Py1,4]+ are composed of a pentagon ring with butyl or ethyl and methyl side chains of similar ionic structure. In [Py1,4]+ the positive charge is localized on the sp3-hybridized nitrogen atom in the center of the tetrahedron and is a non-π-electron system. On the other hand, the cations [BMI]+ and [EMIm]+ are πelectron systems, and the electron is delocalized on the flat aromatic ring π bond. The difference in molecular sizes and the

Figure 7. Comparison of theoretical curves with experimental data: (a) magnitude of impedance and (b) phase of impedance data of Au(111)36 (blue square) and Au(100)37 (red circle) in BMIM-PF6. The blue lines in (a) and (b) represent the magnitude of impedance and the phase response of the two-state model, respectively. The red lines in (a) and (b) represent the magnitude and phase of the impedance response of the three-state model, respectively.

Au(111)36 (blue square) and Au(100)37 (red circle) in BMIMBF4 with the theoretical model. The blue lines in Figure 7(a) and (b) represent the magnitude of impedance and the phase response of the two-state model. The impedance and phase data of Au(111) show a good fit with the two-state model (blue lines). However, Au(100) electrode data fitting suggests that the two-state model (which is represented by black dotted lines) is not sufficient for this system as this model shows the deviation in low CPE regime. There exists an additional state Table 1. Physical Parameters for the Electrode/IL Interfacea figure 7 8

Au(hkl) (111) (100) (111) (111)

ϵH b

11.7 11.7b 12.8b 11.4

τIII H

τIIH

τIH

rH

γ

γc

ω3

ω2

ω1

Rct

726 726 99 174

1.6 127 0.43 3

0.6 -

2.8c 2.8c 2.6c 2.9c

0.84 0.97 0.95 0.95

0.2 0.55 0.23 0.5

0.75 0.6 0.6 0.4

0.25 0.25 0.4 0.6

0.15 -

10 10 20 0.25

Here rH (Angstrom) is the effective thickness of HL corresponding to the radius of ion size of IL; ϵH is the dielectric constant; and Rct is the charge b c transfer resistance in MΩ cm2. The relaxation times τIH, τIIH, and τIII H are in milliseconds (ms). Ref 52. Ref 54. a

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•ILs with cations having a π-electron system, viz., BMI+ and EMIm+, are likely to show more slower relaxation dynamics than the non-π-electron system, viz., [Py1,4], due to stronger adsorption at negative applied potential. Finally, the theory is robust enough to be applied to other classes of ionic liquids.

electronic nature (π or non-π) will affect not only the adsorption but also the electronic interaction in the compact layer. This will have important consequences in deciding the effective capacitance and resistance of the compact layer. The presence of the π-electron system will lead to stronger adsorption with a simultaneous decrease in the thickness of the compact layer. This can be understood as cation reorganization with parallel alignment between the imidazolium ring and electrode surface which also brings them closer in contact. This process can decrease the effective thickness of the compact layer. At negative potential the [BMI]+ ring aligned more parallel to the surface.9 Thus, in the case of the Au(111) system9 a more negative potential of −0.2 V (Figure 7 Au(111) data in Table 1) is applied, and the ring will prefer a more parallel alignment of [BMI]+ than in the case of the Au(100) system where a potential of −0.1 V (Figure 7 Au(100) data in Table 1) is applied. Also the effective resistance of the compact layer in [Py1,4][FAP] will be less than [BMI][PF6] or [EMI][FAP].

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: 91-11-27666646188, 149. Fax: 91-11-27666605. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS R.K. is grateful to DST SERB project grant and University of Delhi for R & D grant.

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CONCLUSIONS We have developed an electric double-layer (EDL) model for the dynamic impedance response of the room-temperature ionic liquids (RTILs). The model accounts for the multistate compact layer to incorporate ion shape asymmetry in RTILs, influence of electrode heterogeneity on such a compact layer, sluggish charge transfer kinetics, and Debye−Falkenhagen relaxation in the diffuse layer region. The ion shape asymmetry causes multistate relaxation admittance in a compact layer on the heterogeneous electrode where the weight and relaxation time of each state depend on molecular orientational properties. We have shown how different molecular states due to ion orientation in the compact part of the double layer influence the dynamics and characteristic time scales. In particular, the following conclusions are drawn from the EIS model proposed and its comparison with recent experimental data: •Our model shows that the single-state compact layer on the heterogeneous electrode has a constant phase element (CPE) behavior. The existence of more than one orientational state in the compact layer can result in oscillation in CPE behavior, i.e., low-frequency camel hump in phase. •The enhanced heterogeneity of the surface can induce multiple substates leading to an average state which can destroy phase modulation due to orientational states of ions. A similar situation can arise due to dilution of RTILs. •Relaxation time of each orientational state in the compact layer can be looked upon as a product of effective resistance and capacitance. •Change in relaxation time of a state or effective resistance shifts the position of the peak in the impedance phase. •The change in capacitance due to orientation is relatively weak as electrode and ion charge separation in OHP is more or less the same. It shows a weak influence in the low-frequency regime. •In the presence of an intermediate state, the phase shows a broadening in the camel hump shape. •The comparisons of theory with data by Pajkossy et al. suggest a possibility of more than two states in the EDL. We show that the modified model with two or three states is sufficient for systems with shape asymmetric cations, and it explains anomalous dynamics in RTILs.

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